Is tidal heating sufficient to explain bloated exoplanets? Consistent calculations accounting for finite initial eccentricity

<p>In close exoplanetary systems, tidal interactions are known to shape the orbital architecture of the system, modify star and planet spins, and have an impact on the internal structure of the bodies through tidal heating. Most stars around which planets have been discovered are low-mass stars and thus feature a magnetised convective envelope, as is also expected in giant gaseous planets like Hot-Jupiter. Tidal flows, and more specifically inertial waves (restored by the Coriolis acceleration, and recently discovered in the Sun) tidally-excited, are the main direct manifestation of tidal interactions in the convective envelopes of these bodies. Furthermore, inertial waves are small-scale waves that are sensitive to nonlinearities, especially in close Hot-Jupiter systems with strong tidal forcing. The nonlinear self-interaction of inertial waves is known to trigger differential rotation in convective shells, as shown in numerical and experimental hydrodynamical studies. Since inertial waves are a key contribution to the tidal dissipation in close star-planet systems, it is essential to have the finest understanding of tidal inertial wave propagation and dissipation in such a complex nonlinear, magnetised, and differentially rotating environment.<br />In this context, we investigate how nonlinearities affect the tidal flow properties, thanks to new 3D hydrodynamic and magneto-hydrodynamic nonlinear simulations of tides, in an adiabatic and incompressible convective shell. First, we show to what extent the emergence of differential rotation is modifying the tidal dissipation rates, prior to linear predictions. In this newly generated zonal flows, nonlinear self-interactions of tidal inertial waves can also trigger different kind of instabilities and resonances between the waves and the background sheared flow, when the tidal forcing is strong enough or the viscosity low enough. These different processes disrupt the energetic exchanges between tidal waves and the background flow, and also further modifies the tidal dissipation rates. Secondly, we present the first non-linear numerical analysis of tidal flows in a magnetised convective shell. One main effect of the magnetic field in our model is to mitigate zonal flows triggered by the nonlinear interaction of inertial waves. The consequences for tidal flows are important, since the installation of zonal flows in nonlinear hydrodynamical simulations is the main cause of significant changes in tidal dissipation and angular momentum exchanges, compared to linear predictions for a uniformly rotating body. </p>

We study how stably stratified or semi-convective layers alter the tidal dissipation rates associated with the generation of internal waves in planetary interiors. We consider if these layers could contribute to the high rates of tidal dissipation observed for Jupiter and Saturn in our solar system. We use an idealized global spherical Boussinesq model to study the influence of stable stratification and semi-convective layers on tidal dissipation rates. We carry out analytical and numerical calculations considering realistic tidal forcing and measure how the viscous and thermal dissipation rates depend on the parameters relating to the internal stratification profile. We find that the strongly frequency-dependent tidal dissipation rate is highly dependent on the parameters relating to the stable stratification, with strong resonant peaks that align with the internal modes of the system. The locations and sizes of these resonances depend on the form and parameters of the stratification, which we explore both analytically and numerically. Our results suggest that stable stratification can significantly enhance the tidal dissipation in particular frequency ranges. Analytical calculations in the low-frequency regime give us scaling laws for the key parameters, including the tidal quality factor due to internal gravity waves. Stably stratified layers can significantly contribute to tidal dissipation in solar and extrasolar giant planets, and we estimate substantial tidal evolution for hot Neptunes. Further investigation is needed to robustly quantify the significance of the contribution in realistic interior models, and to consider the contribution of inertial waves.

The tides of Io

Enceladus's crust as a non-uniform thin shell: II tidal dissipation

A key controller of a planet's rotational evolution, and hence habitability, is tidal dissipation, which on Earth is dominated by the ocean tides. Because exoplanet or deep‐time Earth topographies are unknown, a statistical ensemble is used to constrain possible tidal dissipation rates on an Earth‐like planet. A dedicated tidal model is used together with 120 random continental configurations to simulate Earth's semidiurnal lunar tide. The results show a possible ocean tidal dissipation range spanning 3 orders of magnitude, between 2.3 GWto 1.9 TW (1 TW=1012 W). When model resolution is considered, this compares well with theoretical limits derived for the energetics of Earth's present‐day deep ocean. Consequently, continents exert a fundamental control on tidal dissipation rates and we suggest that plate tectonics on a planet will induce a time‐varying dissipation analogous to Earth's. This will alter rotational periods over millions of years and further complicate the role of tides for planetary evolution.

Tidal heating in icy satellite oceans

Improving tidal modeling for rocky worlds

The motion of two planets around a Sun-like star under the combined effects of mutual interaction and tidal dissipation is investigated. The secular behaviour of the system is analyzed using two different approaches. First, we solve the exact equations of motion through the numerical simulation of the system evolution. In addition to the orbital decay and circularization, we show that the final configuration of the system is affected by the shrink of the inner orbit. Our second approach consist in the analysis of the stationary solutions of mean equations of motion based on a Hamiltonian formalism. We consider the case of a hot super-Earth planet with a more massive outer companion. As a real example, the CoRoT-7 system is analyzed solving the exact and mean equations of motion. The star-planet tidal interaction produces orbital decay and circularization of the orbit of CoRoT-7b. In addition, the long-term tidal evolution is such that the eccentricity of CoRoT-7c is also circularized and a pair of final circular orbits is obtained. A curve in the space of eccentricities can be constructed through the computation of stationary solutions of mean equations including dissipation. The application to CoRoT-7 system shows that the stationary curve agrees with the result of numerical simulations of exact equations. A similar investigation performed in a super-Earth-Jupiter two-planet system shows that the doubly circular state is accelerated when there is a significant orbital migration of the inner planet, in comparison with previous results were migration is neglected.

Residual circulation in the English Channel as a dynamically consistent synthesis of shore-based observations of sea level and currents

Aims. TZ Fornacis is a double-lined eclipsing binary system with similar masses (2.057 ± 0.001 and 1.958 ± 0.001 M⊙) but characterized by very different radii (8.28 ± 0.22 and 3.94 ± 0.17 R⊙). This similarity in terms of mass makes it possible to study the system’s differential stellar evolution as well as some aspects of its tidal evolution. With regard to its orbital elements, it was recently confirmed that its orbit is circular with an orbital period of 75.7 days. The less massive component rotates about 17 times faster than the primary one, which is synchronized with the mean orbital angular velocity. Our main objective in this work is to study both the nuclear and the tidal evolution of the system. Methods. To model the TZ For system, we used the MESA package, computing the grids using the exact observed masses, radii, and effective temperatures as input, and then varying the metallicity, the core overshooting amount, and the mixing-length parameter. A χ2 statistic was used to infer the optimal values of the core overshooting and the mixing-length parameters. The same procedure was used to generate rotating models with the GRANADA code. The respective errors in the average age of TZ For were less than 5%. On the other hand, the differential equations that govern the tidal evolution were integrated using the fifth-order Runge–Kutta method, ith a tolerance of 1 × 10−7. Results. We explored two scenarios regarding the initial eccentricities: a high one (0.30) and a case of an initial circular orbit. A good agreement has been found between the observational values of the eccentricity, synchronism levels, and orbital period with the values predicted by the integration of the tidal evolution equations. The influence of the friction timescale on the evolution of the orbital elements of TZ For is also studied here. The orbital elements most affected by the uncertainties in the friction timescale are the synchronism levels of the two components. On the other hand, we used the properties of the rotating models generated by the GRANADA code as the initial angular velocities instead of using trial values. In this case, comparisons between the theoretical values of the orbital elements and their observed counterparts also lead to a good interagreement.

We provide a generalized discussion of tidal evolution to arbitrary order in the expansion of the gravitational potential between two spherical bodies of any mass ratio. To accurately reproduce the tidal evolution of a system at separations less than 5 times the radius of the larger primary component, the tidal potential due to the presence of a smaller secondary component is expanded in terms of Legendre polynomials to arbitrary order rather than truncated at leading order as is typically done in studies of well-separated system like the Earth and Moon. The equations of tidal evolution including tidal torques, the changes in spin rates of the components, and the change in semimajor axis (orbital separation) are then derived for binary asteroid systems with circular and equatorial mutual orbits. Accounting for higher-order terms in the tidal potential serves to speed up the tidal evolution of the system leading to underestimates in the time rates of change of the spin rates, semimajor axis, and mean motion in the mutual orbit if such corrections are ignored. Special attention is given to the effect of close orbits on the calculation of material properties of the components, in terms of the rigidity and tidal dissipation function, based on the tidal evolution of the system. It is found that accurate determinations of the physical parameters of the system, e.g., densities, sizes, and current separation, are typically more important than accounting for higher-order terms in the potential when calculating material properties. In the scope of the long-term tidal evolution of the semimajor axis and the component spin rates, correcting for close orbits is a small effect, but for an instantaneous rate of change in spin rate, semimajor axis, or mean motion, the close-orbit correction can be on the order of tens of percent. This work has possible implications for the determination of the Roche limit and for spin-state alteration during close flybys.

Volcanic activity at Europa’s seafloor is one of the key questions regarding the habitability of its subsurface ocean. The suitable conditions for hydrothermalism on Europa’s seafloor are conditioned by the heat released from the underlying silicate mantle, supplied by both radiogenic and tidal heating. The orbital resonance between Io, Europa, and Ganymede forces their orbit and maintains non-zero eccentricities. Tidal heating due to eccentricity tides on Io is so extreme that it  produces intense volcanism.  Even though tidal heating  in Europa’s silicate mantle is expected to be much weaker than on Io due to its greater distance to Jupiter,   Běhounková et al. [1] showed that it may still be sufficient to maintain Europa’s mantle in a partially molten state  for several tens to hundred millions of years, particularly during periods of increased eccentricity. Due to inefficient melt transport through the thick lithosphere of Europa [2], melt produced during periods of enhanced eccentricity may accumulate and in turn affect the tidal heating, as it is the case for Io, implying a possible runaway melt process in the silicate interior of Europa. In this context, the goal of this study is to evaluate the effect of melt accumulation on the tidal heat production of Europa’s silicate mantle. For that purpose, we follow the approach developed to model the solid tides in Io’s partially molten interior [3], taking into account the effect of melt on the viscoelastic properties of the mantle. We adapt it to the context of Europa, corresponding to a deeper and thinner asthenosphere than on Io. We show that, whatever the partially molten layer thickness, melt accumulation increases tidal heat production and tidal dissipation even exceeds radiogenic heating. For equivalent volume of accumulated melt, the thinner the layer, the more pronounced this effect is. Our results show that the accumulation of melt, over timescales consistent with the 3D model prediction of Běhounková et al. [1], may significantly affect the tidal dissipation amplitude and its pattern. The potential presence of such melt accumulations may be tested by future measurements by Europa Clipper and JUICE from the combined analysis of gravimetric, altimetric and magnetic data, which might reveal long-wavelength anomalies which could be confronted to our model prediction. [1] Běhounková et al., GRL, 48(3), e2020GL090077 (2021),  [2] Bland and Elder, GRL, 49(5), e2021GL096939 (2022). [3] Kervazo et al., A&A, 650, A72 (2021) 

We study the susceptibility of extrasolar Earth-like planets to tidal dissipation by varying orbital, rheological, and heat transfer parameters. We employ a three-dimensional numerical method solving the coupled problem of mantle convection and tidal dissipation. A reference model mimicking a plate tectonic regime and reproducing Earth's present-day heat output is considered. Four other models representing less efficient heat transfer regimes are also investigated. For these five initial models, we determine the orbital configurations under which a positive feedback between tidal dissipation and temperature evolution leads to a thermal runaway. In order to describe the occurrence of thermal runaways, we develop a scaling that relates the global dissipated power to a characteristic temperature and to the orbital parameters. For all numerical experiments sharing the same initial temperature conditions, we show that the reciprocal value of the runaway timescale depends linearly on the global dissipated power at the beginning of the simulation. In the plate tectonic-like regime, Earth-like planets in the habitable zone (HZ) of 0.1 M☉ stars experience thermal runaways for 1:1 spin–orbit resonance if the eccentricity is sufficiently high (e>0.02 at a 4 day period, e>0.2 at a 10 day period). For less efficient convective regimes, runaways are obtained for eccentricities as low as ∼0.004 at the inner limit of the HZ. In the case of 3:2 spin–orbit resonance, the occurrence of thermal runaways is independent of eccentricity and is predicted for orbital periods lower than 12 days. For less efficient convective regimes, runaways may occur at larger orbital periods potentially affecting the HZ of stars with a mass up to 0.4 M☉. Whatever the convective regime and spin–orbit resonance, tidal heating within Earth-like planets orbiting in the HZ of stars more massive than 0.5 M☉ is not significant.

Context. Tidal dissipation in planets and in stars is one of the key physical\nmechanisms that drive the evolution of planetary systems.\n Aims. Tidal dissipation properties are intrisically linked to the internal\nstructure and the rheology of studied celestial bodies. The resulting\ndependence of the dissipation upon the tidal frequency is strongly different in\nthe cases of solids and fluids.\n Methods. We compute the tidal evolution of a two-body coplanar system, using\nthe tidal quality factor's frequency-dependencies appropriate to rocks and to\nconvective fluids.\n Results. The ensuing orbital dynamics comes out smooth or strongly erratic,\ndependent on how the tidal dissipation depends upon frequency.\n Conclusions. We demonstrate the strong impact of the internal structure and\nof the rheology of the central body on the orbital evolution of the tidal\nperturber. A smooth frequency-dependence of the tidal dissipation renders a\nsmooth orbital evolution while a peaked dissipation can furnish erratic orbital\nbehaviour.\n

Context. Binary stars cannot be fully understood without assessing the interaction effects between the two components and the impact of these effects on observational diagnostics. Periastron brightening events, also known as the heartbeat phenomenon, are a clear manifestation of this type of interaction. Aims. We aim to explore the role of tidal shear energy dissipation in stars undergoing periastron brightening events. Methods. We performed a computation from first principles that uses a quasi-hydrodynamic Lagrangian scheme to simultaneously solve the orbital motion and the equations of motion of a 3D grid of volume elements covering the inner, rigidly rotating “core” of a tidally perturbed star. The equations of motion include the gravitational acceleration of both stars, the centrifugal, Coriolis, gas pressure accelerations, and viscous coupling between volume elements. The method is illustrated for a grid of model binary systems with a 10 M⊙ primary that is perturbed by a 6.97 M⊙ companion in eccentric orbits (e = 0 − 0.7). The model is then applied to the heartbeat star MACHO 80.7443.1718. Results. We find an increase by factors ∼10−6–10−3 in tidal shear energy dissipation at periastron, consistent with the majority of observed heartbeat stars. The magnitude of the periastron effect correlates with the degree of departure from synchronicity: stars rotating much faster or much slower than the synchronous rate at periastron present the strongest effect. We confirm that for eccentricities ≤0.3, pseudo-synchronization occurs for 0.8 < ω/Ωave < 1, where Ωave is the average orbital angular velocity. The minimum energy rotation rate (pseudo-synchronism) for e = 0.5 and 0.7 occurs for 1.0 < ω/Ωave < 1.15. The tidal shear energy dissipation model reproduces from first principles the ∼23% maximum brightness enhancement at periastron of MACHO 80.7443.1718. Conclusions. Our results suggest that the magnitude and shape of the heartbeat signal may serve as diagnostics for the internal stellar rotation and turbulent viscosity values.